1. Field of Invention
The present invention relates generally to a laser vibrometry method and system and, more specifically, to a method and system for processing laser vibrometry data employing Bayesian statistical processing techniques.
2. Description of the Related Art
Laser vibrometry (LV) provides a sensitive non-contact means of measuring vibrations of objects. In a LV measurement, a laser beam illuminates an object of interest, which may be stationary or in motion, and the returned scattered light is mixed with a local oscillator derived from the same laser or a different laser frequency-referenced to the first. The instantaneous beat frequency provides a measurement of the surface velocity that may be extracted from the time-series data by signal processing (for example, a Fourier transform). The time evolution of the beat frequency then contains information about the state of motion (vibration or other time-varying velocity) of the target, which can be extracted by further processing. Traditional methods for processing LV data include the FM discriminator method, spectrogram processing (a Fourier method), and time-frequency distributions. See, e.g., A. L. Kachelmyer and K. I. Schultz, xe2x80x9cLaser vibration sensing,xe2x80x9d Linc. Lab. J. 8(1), pp. 3-28, 1995 and T. D. Cole and A. S. El-Dinary, xe2x80x9cEstimation of target vibration spectra from laser radar backscatter using time-frequency distributions,xe2x80x9d in Applied Laser Radar Technology, Proc. SPIE, G. W. Kamerman and W. E. Keicher, eds., vol. 1936, pp. 90-103, SPIE, (Bellingham, Wash.), 1993, both of which are incorporated herein by reference.
Since the development of the Fast Fourier Transform (FFT), discrete Fourier transforms have become for many the first and last word in analyzing the frequency content of a signal. In many circumstances this use of discrete transforms is quite well justified. There are, however, many important instances where the FFT-based approach to spectral estimation is not optimal. One such example is that of short time series. The resolution of the discrete Fourier transforms is determined by the sample duration. When this interval only covers a few periods of the frequency of interest, it can be difficult to obtain good frequency estimates. This difficulty is further exacerbated by the leakage of the spectral response of the data windowing function into the spectral range of the signal. Even if no window function is explicitly applied (a questionable practice in any event) there is an implicit windowing of the data that can not be avoided.
In order to be of utility for laser vibration sensing, a signal processing method must operate well in the presence of noise, be robust to speckle broadening and laser linewidth, and be computationally efficient. It should also yield useful spectra in a small number of vibrational periods, especially when the vibrational period is long or if the measurement time is limited. This latter requirement poses a significant constraint on FFT approaches because the frequency resolution is roughly equal to the inverse of the measurement time. Accordingly, there is a need for a signal processing approach that provides improved performance and resolution for laser vibration sensing.
According to the present invention, an approach to laser vibrometry data analysis based on statistical inference is employed. Generally, the methods of the present invention (broadly labeled as Maximum Entropy) depart from traditional data-processing approaches in favor of the modeling of experiments. This distinction is more than merely pedantic and allows not only for a sound theoretical basis for estimation of various parameters of interest but also for assignment of confidence levels to these parameter estimates as well as for making relative quantitative assessments between competing models as to which is most consistent with the data.
While the traditional approach to data analysis involves working xe2x80x9cbackwardsxe2x80x9d from the measured data to determine the parameters of interest in the model, in an exemplary preferred embodiment of the present invention, data analysis based on Bayesian statistical inference works xe2x80x9cforwardsxe2x80x9d from the model to determine the model most statistically consistent with the data. Thus, the question is asked, xe2x80x9cHow likely is it that the observed data is a consequence of the model?xe2x80x9d
The statistical framework of the present invention allows for much more than the fitting of parameters in the model to match the data. In particular, often there are parameters in models that are essential for describing the data but otherwise do not contain physically relevant information. In a standard fitting approach, these uninteresting parameters would still nonetheless have to be included in along with xe2x80x9cinterestingxe2x80x9d parameters. Moreover, it is not uncommon for these uninteresting parameters to out-number the interesting parameters, making the fitting procedure a great deal more work than if one could somehow consider only the physically relevant parameters. According to the statistical approach of the present invention, it is possible to marginalize (integrate out) these uninteresting parameters (often called nuisance parameters) leaving only the physically important parameters behind. Effectively this facilitates determination of the values of the relevant parameters most consistent with the data knowing that the nuisance parameters will take on whatever values necessary to be consistent with the data. A germane example of nuisance parameters is the phase and quadrature amplitudes of a sinusoidal signal; very often only the frequencies of vibration are of interest. According to the present invention, Bayesian methods provide the framework to consider the important parameters of models of systems under observation while (effectively) ignoring the unimportant parameters. Because of the optimal use of prior knowledge about the laser vibrometry signal provided by the Bayesian statistical signal processing method and system of the present invention, the frequencies can be determined with much greater precision and greater noise immunity than using Fourier- or time-frequency-based approaches. Furthermore, the Bayesian approach of the present invention provides superior performance when the data extends over a small number of vibration periods. The potential advantage of this technique is several orders of magnitude improvement in the precision of the determination of vibrational frequencies, depending upon experimental conditions. An improvement of about a factor of 50 in the precision of the determination of vibrational frequencies has been observed employing the techniques of the present invention.
In accordance with one embodiment of the present invention, a laser vibrometry method includes the steps of: employing a laser to generate laser vibrometry data for a system under observation (e.g., a reflective target); and performing Bayesian parameter estimation calculations for a mathematical model of the system under observation, the laser vibrometry data and prior information to generate estimations of parameters of the mathematical model. In a preferred embodiment, the laser vibrometry method farther includes the steps of evaluating the estimations of parameters; and, if desired or necessary, repeating the step of performing Bayesian parameter estimation calculations with a different model and/or different laser vibrometry data. The laser vibrometry data is, for example, continuous wave (CW) laser vibrometry data, dual pulse coherent laser vibrometry data, or vibrational imaging data. By way of example, the prior information is noise information such as noise information pertaining to a signal-to-noise ratio, or noise information that is assigned a least informative probability density. In a preferred embodiment, the step of evaluating parameter estimation results includes evaluating vibration frequencies. In a preferred embodiment, the step of evaluating parameter estimation results includes evaluating variances of parameter estimates. In a preferred embodiment, the step of evaluating parameter estimation results includes evaluating higher order moments.
In accordance with another embodiment of the present invention, a laser vibrometry method includes the steps of: creating one or more mathematical models of a system under observation; employing a laser to generate laser vibrometry data for the system under observation; and processing the one or more mathematical models, the laser vibrometry data and prior information employing a statistical signal processing technique to generate estimates of parameters for the one or more mathematical models. In a preferred embodiment, the laser vibrometry method further includes the steps of: performing an evaluation of the parameter estimations; and, if results of the evaluation are determined not to be satisfactory, repeating the processing step with a different model and/or different laser vibrometry data. In a preferred embodiment, the statistical signal processing technique employs Bayesian decision theory. In a preferred embodiment, the statistical signal processing technique employs a Bayesian statistical inference technique. In a preferred embodiment, the statistical signal processing technique employs a subjectivist inference technique.
In accordance with another embodiment of the present invention, a system for processing laser vibrometry data includes: a processor configured to employ a statistical signal processing technique to process one or more mathematical models, laser vibrometry data for a system under observation, and prior information to generate estimates of parameters for the one or more mathematical models. In a preferred embodiment, the statistical signal processing technique comprises a Bayesian processing technique. In a preferred embodiment, the one or more mathematical models are models of the system under observation. In a preferred embodiment, the processor is configured to process parameter estimation results to facilitate an evaluation of hypotheses about parameters of a process generating the laser vibrometry data.
The method and system of the present invention are suitable for commercial industrial laser vibrometry, laser velocimetry devices (e.g., fiber-optic blood flow measurement devices), coherent laser processing, and vibrational imaging. Measurements of vibrations of objects can be used in industrial quality control and wear monitoring as well as in the analysis of the vibrational characteristics of objects. Other applications include failure analysis of mechanical devices, identification of objects/machinery, and laser microphones.
The above described and many other features and attendant advantages of the present invention will become apparent as the invention becomes better understood by reference to the following detailed description when considered in conjunction with the accompanying drawings.